In the paper, some consensus problems of multi-agent system controlled by edge event-triggered strategy and state-constraint impulsive are taken into account. For the state-constraint impulsive protocol, two types of control protocols which contain input saturation and double actuator saturation are put forward. With regard to the edge event-triggered control, we propose the rule of it and let the time of the edge event-triggered be impulsive time to avoid the Zeno-behavior. Then, compared to the control method of others, we can greatly reduce the cost in the process of exchanging information. Next, some sufficient conditions of the system are required to reach consensus. In the end, a few examples are exploited for testing and checking the theoretical analyses.